In the present study, a continuous-based thermo-electromechanic model has been developed by the Kirchhoff plate’s theory and the modified flexoelectric theory in order to study the size-dependent nonlinear free vibration of functionally graded flexoelectric nano-plate under the magnetic field. Using the Hamilton’s principle and variation method, the nonlinear governing differential equations of the nano-plate and their associated boundary conditions have been extracted and the governing equations solved by using Galerkin’s and perturbation methods. The electromechanical coupling (electromechanical stress) in the internal energy function causes nonlinearity in the governing equations. The applied magnetic field is a type of external static field along with the nano-plate thickness. The natural frequencies and related mode shapes have been determined in two modes of direct and inverse flexoelectric effects. Also, the effects of such factors as length scale parameters, geometric parameters, thermal, magnetic and electrical loadings were investigated. In the presence of flexoelectric effect, the results showed that the dependence of electromechanical behavior of the structure on size is found to be significant in nanoscales. Regarding the application of this type of nano-plate in the oscillators and considering the flexoelectric effect, the applied potential difference can play an important role in adjusting and controlling the frequency.

中文翻译：

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尺寸相关的功能梯度柔性纳米板暴露磁场的非线性热机电振动分析

在本研究中，通过基尔霍夫板理论和改进的柔性电理论建立了基于连续的热机电模型，以研究磁场作用下功能梯度柔性纳米板的尺寸依赖性非线性自由振动。利用汉密尔顿原理和变分法，提取了纳米板的非线性控制微分方程及其相关的边界条件，并通过伽勒金法和摄动法求解了该控制方程。内部能量函数中的机电耦合（机电应力）导致控制方程非线性。所施加的磁场是一种外部静电场，它与纳米板的厚度一起。固有频率和相关的模态形状已通过正和反柔电效应的两种模式确定。此外，还研究了诸如长度比例参数，几何参数，热，磁和电负载等因素的影响。在存在柔电效应的情况下，结果表明，该结构的机电行为对尺寸的依赖性在纳米级上是显着的。关于这种类型的纳米板在振荡器中的应用并考虑柔电效应，所施加的电势差可以在调节和控制频率中起重要作用。研究了磁和电负载。在存在柔电效应的情况下，结果表明，该结构的机电行为对尺寸的依赖性在纳米级上是显着的。关于这种类型的纳米板在振荡器中的应用并考虑柔电效应，所施加的电势差可以在调节和控制频率中起重要作用。研究了磁和电负载。在存在柔电效应的情况下，结果表明，该结构的机电行为对尺寸的依赖性在纳米级上是显着的。关于这种类型的纳米板在振荡器中的应用并考虑柔电效应，所施加的电势差可以在调节和控制频率中起重要作用。